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The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It can be written as: a² + b² = c², where c is the hypotenuse and a and b are the other two sides. This theorem is fundamental in geometry and has applications in architecture, navigation, and more.